Tools with adjustable interchangeable cartridges can relatively easily be adapted to different workpiece dimensions and machining tasks and can therefore be utilized more universally than special tools with nonadjustable inserts. This advantage has mainly an effect on small and medium workpiece series such that few universal tools are sufficient and that a plurality of special tools need not be manufactured and kept in storage. On the other hand, the adjustability and the thus needed division of the tool into several parts causes difficulties with respect to stability and strength, which could often not be satisfactorily solved with the presently known jointed structures in order to meet high performance requirements. Thus, it has already been suggested to adjust the interchangeable cartridge with the help of an adjusting screw guided in advance direction of the cutting head and resting loosely against an edge of the interchangeable cartridge to the desired cutting depth, and to thereafter clamp same on the cutting head with the help of a fastening screw. A tilting moment (or torque) is applied during the machining operation around the rearward bearing edge of the interchangeable cartridge at the cutting head, which bearing edge is adjacent to the main cutting edge, through the insert onto the interchangeable cartridge. The tilting moment must be completely absorbed by the fastening screw since the adjustment screw does not contribute anything toward absorbing this tilting moment. One must add that in the case of a rotating tool the centrifugal forces acting on the interchangeable cartridge of the known tool also must be absorbed exclusively by the fastening screw. Both types of stress reduce the stability particularly at the high speeds desired today.